[{"abstract":[{"lang":"eng","text":"In the class of strictly convex smooth boundaries each of which has no strip around its boundary foliated by invariant curves, we prove that the Taylor coefficients of the “normalized” Mather’s β-function are invariant under C∞-conjugacies. In contrast, we prove that any two elliptic billiard maps are C0-conjugate near their respective boundaries, and C∞-conjugate, near the boundary and away from a line passing through the center of the underlying ellipse. We also prove that, if the billiard maps corresponding to two ellipses are topologically conjugate, then the two ellipses are similar."}],"ec_funded":1,"day":"03","publication_status":"published","oa_version":"Preprint","page":"525-537","date_updated":"2023-08-04T08:59:14Z","status":"public","main_file_link":[{"open_access":"1","url":"https://doi.org/10.48550/arXiv.2105.14640"}],"title":"On some invariants of Birkhoff billiards under conjugacy","external_id":{"arxiv":["2105.14640"],"isi":["000865267300002"]},"related_material":{"link":[{"url":"https://doi.org/10.1134/s1560354722060107","relation":"erratum"}]},"publication":"Regular and Chaotic Dynamics","citation":{"chicago":"Koudjinan, Edmond, and Vadim Kaloshin. “On Some Invariants of Birkhoff Billiards under Conjugacy.” <i>Regular and Chaotic Dynamics</i>. Springer Nature, 2022. <a href=\"https://doi.org/10.1134/S1560354722050021\">https://doi.org/10.1134/S1560354722050021</a>.","ieee":"E. Koudjinan and V. Kaloshin, “On some invariants of Birkhoff billiards under conjugacy,” <i>Regular and Chaotic Dynamics</i>, vol. 27, no. 6. Springer Nature, pp. 525–537, 2022.","apa":"Koudjinan, E., &#38; Kaloshin, V. (2022). On some invariants of Birkhoff billiards under conjugacy. <i>Regular and Chaotic Dynamics</i>. Springer Nature. <a href=\"https://doi.org/10.1134/S1560354722050021\">https://doi.org/10.1134/S1560354722050021</a>","mla":"Koudjinan, Edmond, and Vadim Kaloshin. “On Some Invariants of Birkhoff Billiards under Conjugacy.” <i>Regular and Chaotic Dynamics</i>, vol. 27, no. 6, Springer Nature, 2022, pp. 525–37, doi:<a href=\"https://doi.org/10.1134/S1560354722050021\">10.1134/S1560354722050021</a>.","short":"E. Koudjinan, V. Kaloshin, Regular and Chaotic Dynamics 27 (2022) 525–537.","ista":"Koudjinan E, Kaloshin V. 2022. On some invariants of Birkhoff billiards under conjugacy. Regular and Chaotic Dynamics. 27(6), 525–537.","ama":"Koudjinan E, Kaloshin V. On some invariants of Birkhoff billiards under conjugacy. <i>Regular and Chaotic Dynamics</i>. 2022;27(6):525-537. doi:<a href=\"https://doi.org/10.1134/S1560354722050021\">10.1134/S1560354722050021</a>"},"arxiv":1,"project":[{"_id":"9B8B92DE-BA93-11EA-9121-9846C619BF3A","name":"Spectral rigidity and integrability for billiards and geodesic flows","grant_number":"885707","call_identifier":"H2020"}],"scopus_import":"1","issue":"6","_id":"12145","oa":1,"author":[{"last_name":"Koudjinan","id":"52DF3E68-AEFA-11EA-95A4-124A3DDC885E","orcid":"0000-0003-2640-4049","first_name":"Edmond","full_name":"Koudjinan, Edmond"},{"orcid":"0000-0002-6051-2628","id":"FE553552-CDE8-11E9-B324-C0EBE5697425","last_name":"Kaloshin","full_name":"Kaloshin, Vadim","first_name":"Vadim"}],"date_published":"2022-10-03T00:00:00Z","year":"2022","publisher":"Springer Nature","doi":"10.1134/S1560354722050021","language":[{"iso":"eng"}],"type":"journal_article","publication_identifier":{"issn":["1560-3547"],"eissn":["1468-4845"]},"quality_controlled":"1","month":"10","article_type":"original","volume":27,"isi":1,"acknowledgement":"We are grateful to the anonymous referees for their careful reading and valuable remarks and\r\ncomments which helped to improve the paper significantly. We gratefully acknowledge support from the European Research Council (ERC) through the Advanced Grant “SPERIG” (#885707).","department":[{"_id":"VaKa"}],"date_created":"2023-01-12T12:06:49Z","intvolume":"        27","keyword":["Mechanical Engineering","Applied Mathematics","Mathematical Physics","Modeling and Simulation","Statistical and Nonlinear Physics","Mathematics (miscellaneous)"],"user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","article_processing_charge":"No"},{"main_file_link":[{"url":"https://arxiv.org/abs/2010.13243","open_access":"1"}],"external_id":{"isi":["000614454700004"],"arxiv":["2010.13243"]},"title":"V.I. Arnold's ''Global'' KAM theorem and geometric measure estimates","publication":"Regular and Chaotic Dynamics","arxiv":1,"citation":{"ama":"Chierchia L, Koudjinan E. V.I. Arnold’s “‘Global’” KAM theorem and geometric measure estimates. <i>Regular and Chaotic Dynamics</i>. 2021;26(1):61-88. doi:<a href=\"https://doi.org/10.1134/S1560354721010044\">10.1134/S1560354721010044</a>","apa":"Chierchia, L., &#38; Koudjinan, E. (2021). V.I. Arnold’s “‘Global’” KAM theorem and geometric measure estimates. <i>Regular and Chaotic Dynamics</i>. Springer Nature. <a href=\"https://doi.org/10.1134/S1560354721010044\">https://doi.org/10.1134/S1560354721010044</a>","mla":"Chierchia, Luigi, and Edmond Koudjinan. “V.I. Arnold’s ‘“Global”’ KAM Theorem and Geometric Measure Estimates.” <i>Regular and Chaotic Dynamics</i>, vol. 26, no. 1, Springer Nature, 2021, pp. 61–88, doi:<a href=\"https://doi.org/10.1134/S1560354721010044\">10.1134/S1560354721010044</a>.","short":"L. Chierchia, E. Koudjinan, Regular and Chaotic Dynamics 26 (2021) 61–88.","ista":"Chierchia L, Koudjinan E. 2021. V.I. Arnold’s ‘“Global”’ KAM theorem and geometric measure estimates. Regular and Chaotic Dynamics. 26(1), 61–88.","ieee":"L. Chierchia and E. Koudjinan, “V.I. Arnold’s ‘“Global”’ KAM theorem and geometric measure estimates,” <i>Regular and Chaotic Dynamics</i>, vol. 26, no. 1. Springer Nature, pp. 61–88, 2021.","chicago":"Chierchia, Luigi, and Edmond Koudjinan. “V.I. Arnold’s ‘“Global”’ KAM Theorem and Geometric Measure Estimates.” <i>Regular and Chaotic Dynamics</i>. Springer Nature, 2021. <a href=\"https://doi.org/10.1134/S1560354721010044\">https://doi.org/10.1134/S1560354721010044</a>."},"issue":"1","scopus_import":"1","abstract":[{"text":"This paper continues the discussion started in [CK19] concerning Arnold's legacy on classical KAM theory and (some of) its modern developments. We prove a detailed and explicit `global' Arnold's KAM Theorem, which yields, in particular, the Whitney conjugacy of a non{degenerate, real{analytic, nearly-integrable Hamiltonian system to an integrable system on a closed, nowhere dense, positive measure subset of the phase space. Detailed measure estimates on the Kolmogorov's set are provided in the case the phase space is: (A) a uniform neighbourhood of an arbitrary (bounded) set times the d-torus and (B) a domain with C2 boundary times the d-torus. All constants are explicitly given.","lang":"eng"}],"page":"61-88","oa_version":"Preprint","publication_status":"published","day":"03","date_updated":"2023-08-07T13:37:27Z","status":"public","quality_controlled":"1","publication_identifier":{"issn":["1560-3547"]},"month":"02","volume":26,"article_type":"original","isi":1,"department":[{"_id":"VaKa"}],"date_created":"2020-10-21T14:56:47Z","keyword":["Nearly{integrable Hamiltonian systems","perturbation theory","KAM Theory","Arnold's scheme","Kolmogorov's set","primary invariant tori","Lagrangian tori","measure estimates","small divisors","integrability on nowhere dense sets","Diophantine frequencies."],"intvolume":"        26","ddc":["515"],"article_processing_charge":"No","user_id":"4359f0d1-fa6c-11eb-b949-802e58b17ae8","oa":1,"_id":"8689","author":[{"first_name":"Luigi","full_name":"Chierchia, Luigi","last_name":"Chierchia"},{"last_name":"Koudjinan","id":"52DF3E68-AEFA-11EA-95A4-124A3DDC885E","orcid":"0000-0003-2640-4049","first_name":"Edmond","full_name":"Koudjinan, Edmond"}],"date_published":"2021-02-03T00:00:00Z","publisher":"Springer Nature","year":"2021","language":[{"iso":"eng"}],"doi":"10.1134/S1560354721010044","type":"journal_article"},{"file":[{"relation":"main_file","content_type":"application/pdf","access_level":"open_access","creator":"ekoudjin","date_updated":"2021-05-30T13:57:37Z","checksum":"b281b5c2e3e90de0646c3eafcb2c6c25","date_created":"2021-05-30T13:57:37Z","file_id":"9436","file_size":353431,"file_name":"CoExistence 2&3 caustics 3_17_6_2_3.pdf"}],"title":"Non co-preservation of the 1/2 and  1/(2l+1)-rational caustics along deformations of circles","has_accepted_license":"1","ddc":["500"],"user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","article_processing_charge":"No","file_date_updated":"2021-05-30T13:57:37Z","citation":{"ama":"Kaloshin V, Koudjinan E. Non co-preservation of the 1/2 and  1/(2l+1)-rational caustics along deformations of circles. 2021.","apa":"Kaloshin, V., &#38; Koudjinan, E. (2021). Non co-preservation of the 1/2 and  1/(2l+1)-rational caustics along deformations of circles.","ista":"Kaloshin V, Koudjinan E. 2021. Non co-preservation of the 1/2 and  1/(2l+1)-rational caustics along deformations of circles.","mla":"Kaloshin, Vadim, and Edmond Koudjinan. <i>Non Co-Preservation of the 1/2 and  1/(2l+1)-Rational Caustics along Deformations of Circles</i>. 2021.","short":"V. Kaloshin, E. Koudjinan, (2021).","ieee":"V. Kaloshin and E. Koudjinan, “Non co-preservation of the 1/2 and  1/(2l+1)-rational caustics along deformations of circles.” 2021.","chicago":"Kaloshin, Vadim, and Edmond Koudjinan. “Non Co-Preservation of the 1/2 and  1/(2l+1)-Rational Caustics along Deformations of Circles,” 2021."},"department":[{"_id":"VaKa"}],"date_created":"2021-05-30T13:58:13Z","abstract":[{"text":"For any given positive integer l, we prove that every plane deformation of a circlewhich preserves the 1/2and 1/ (2l + 1) -rational caustics is trivial i.e. the deformationconsists only of similarities (rescalings and isometries).","lang":"eng"}],"date_published":"2021-01-01T00:00:00Z","oa_version":"Submitted Version","oa":1,"_id":"9435","author":[{"first_name":"Vadim","full_name":"Kaloshin, Vadim","orcid":"0000-0002-6051-2628","id":"FE553552-CDE8-11E9-B324-C0EBE5697425","last_name":"Kaloshin"},{"orcid":"0000-0003-2640-4049","last_name":"Koudjinan","id":"52DF3E68-AEFA-11EA-95A4-124A3DDC885E","full_name":"Koudjinan, Edmond","first_name":"Edmond"}],"type":"preprint","status":"public","date_updated":"2021-06-01T09:10:22Z","year":"2021","language":[{"iso":"eng"}]},{"type":"journal_article","year":"2020","publisher":"Elsevier","doi":"10.1016/j.jde.2020.03.044","language":[{"iso":"eng"}],"date_published":"2020-09-05T00:00:00Z","_id":"8691","oa":1,"author":[{"orcid":"0000-0003-2640-4049","last_name":"Koudjinan","id":"52DF3E68-AEFA-11EA-95A4-124A3DDC885E","full_name":"Koudjinan, Edmond","first_name":"Edmond"}],"intvolume":"       269","keyword":["Analysis"],"extern":"1","article_processing_charge":"No","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","date_created":"2020-10-21T15:03:05Z","article_type":"original","volume":269,"publication_identifier":{"issn":["0022-0396"]},"quality_controlled":"1","month":"09","status":"public","date_updated":"2021-01-12T08:20:33Z","abstract":[{"lang":"eng","text":"Given l>2ν>2d≥4, we prove the persistence of a Cantor--family of KAM tori of measure O(ε1/2−ν/l) for any non--degenerate nearly integrable Hamiltonian system of class Cl(D×Td), where D⊂Rd is a bounded domain, provided that the size ε of the perturbation is sufficiently small. This extends a result by D. Salamon in \\cite{salamon2004kolmogorov} according to which we do have the persistence of a single KAM torus in the same framework. Moreover, it is well--known that, for the persistence of a single torus, the regularity assumption can not be improved."}],"day":"05","publication_status":"published","page":"4720-4750","oa_version":"Preprint","issue":"6","arxiv":1,"citation":{"ieee":"E. Koudjinan, “A KAM theorem for finitely differentiable Hamiltonian systems,” <i>Journal of Differential Equations</i>, vol. 269, no. 6. Elsevier, pp. 4720–4750, 2020.","chicago":"Koudjinan, Edmond. “A KAM Theorem for Finitely Differentiable Hamiltonian Systems.” <i>Journal of Differential Equations</i>. Elsevier, 2020. <a href=\"https://doi.org/10.1016/j.jde.2020.03.044\">https://doi.org/10.1016/j.jde.2020.03.044</a>.","ama":"Koudjinan E. A KAM theorem for finitely differentiable Hamiltonian systems. <i>Journal of Differential Equations</i>. 2020;269(6):4720-4750. doi:<a href=\"https://doi.org/10.1016/j.jde.2020.03.044\">10.1016/j.jde.2020.03.044</a>","short":"E. Koudjinan, Journal of Differential Equations 269 (2020) 4720–4750.","apa":"Koudjinan, E. (2020). A KAM theorem for finitely differentiable Hamiltonian systems. <i>Journal of Differential Equations</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.jde.2020.03.044\">https://doi.org/10.1016/j.jde.2020.03.044</a>","mla":"Koudjinan, Edmond. “A KAM Theorem for Finitely Differentiable Hamiltonian Systems.” <i>Journal of Differential Equations</i>, vol. 269, no. 6, Elsevier, 2020, pp. 4720–50, doi:<a href=\"https://doi.org/10.1016/j.jde.2020.03.044\">10.1016/j.jde.2020.03.044</a>.","ista":"Koudjinan E. 2020. A KAM theorem for finitely differentiable Hamiltonian systems. Journal of Differential Equations. 269(6), 4720–4750."},"title":"A KAM theorem for finitely differentiable Hamiltonian systems","external_id":{"arxiv":["1909.04099"]},"publication":"Journal of Differential Equations","main_file_link":[{"url":"https://arxiv.org/abs/1909.04099","open_access":"1"}]},{"oa":1,"_id":"8694","author":[{"first_name":"Ali","full_name":"Golmakani, Ali","last_name":"Golmakani"},{"full_name":"Koudjinan, Edmond","first_name":"Edmond","orcid":"0000-0003-2640-4049","last_name":"Koudjinan","id":"52DF3E68-AEFA-11EA-95A4-124A3DDC885E"},{"last_name":"Luzzatto","first_name":"Stefano","full_name":"Luzzatto, Stefano"},{"full_name":"Pilarczyk, Pawel","first_name":"Pawel","last_name":"Pilarczyk"}],"date_published":"2020-07-31T00:00:00Z","publisher":"AIP","year":"2020","language":[{"iso":"eng"}],"doi":"10.1063/5.0012822","type":"journal_article","quality_controlled":"1","month":"07","article_type":"original","volume":30,"date_created":"2020-10-21T15:43:05Z","article_number":"073143","intvolume":"        30","extern":"1","article_processing_charge":"No","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","abstract":[{"text":"We develop algorithms and techniques to compute rigorous bounds for finite pieces of orbits of the critical points, for intervals of parameter values, in the quadratic family of one-dimensional maps fa(x)=a−x2. We illustrate the effectiveness of our approach by constructing a dynamically defined partition 𝒫 of the parameter interval Ω=[1.4,2] into almost 4×106 subintervals, for each of which we compute to high precision the orbits of the critical points up to some time N and other dynamically relevant quantities, several of which can vary greatly, possibly spanning several orders of magnitude. We also subdivide 𝒫 into a family 𝒫+ of intervals, which we call stochastic intervals, and a family 𝒫− of intervals, which we call regular intervals. We numerically prove that each interval ω∈𝒫+ has an escape time, which roughly means that some iterate of the critical point taken over all the parameters in ω has considerable width in the phase space. This suggests, in turn, that most parameters belonging to the intervals in 𝒫+ are stochastic and most parameters belonging to the intervals in 𝒫− are regular, thus the names. We prove that the intervals in 𝒫+ occupy almost 90% of the total measure of Ω. The software and the data are freely available at http://www.pawelpilarczyk.com/quadr/, and a web page is provided for carrying out the calculations. The ideas and procedures can be easily generalized to apply to other parameterized families of dynamical systems.","lang":"eng"}],"publication_status":"published","oa_version":"Preprint","day":"31","date_updated":"2021-01-12T08:20:34Z","status":"public","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2004.13444"}],"external_id":{"arxiv":["2004.13444"]},"title":"Rigorous numerics for critical orbits in the quadratic family","publication":"Chaos","arxiv":1,"citation":{"ieee":"A. Golmakani, E. Koudjinan, S. Luzzatto, and P. Pilarczyk, “Rigorous numerics for critical orbits in the quadratic family,” <i>Chaos</i>, vol. 30, no. 7. AIP, 2020.","chicago":"Golmakani, Ali, Edmond Koudjinan, Stefano Luzzatto, and Pawel Pilarczyk. “Rigorous Numerics for Critical Orbits in the Quadratic Family.” <i>Chaos</i>. AIP, 2020. <a href=\"https://doi.org/10.1063/5.0012822\">https://doi.org/10.1063/5.0012822</a>.","ama":"Golmakani A, Koudjinan E, Luzzatto S, Pilarczyk P. Rigorous numerics for critical orbits in the quadratic family. <i>Chaos</i>. 2020;30(7). doi:<a href=\"https://doi.org/10.1063/5.0012822\">10.1063/5.0012822</a>","apa":"Golmakani, A., Koudjinan, E., Luzzatto, S., &#38; Pilarczyk, P. (2020). Rigorous numerics for critical orbits in the quadratic family. <i>Chaos</i>. AIP. <a href=\"https://doi.org/10.1063/5.0012822\">https://doi.org/10.1063/5.0012822</a>","ista":"Golmakani A, Koudjinan E, Luzzatto S, Pilarczyk P. 2020. Rigorous numerics for critical orbits in the quadratic family. Chaos. 30(7), 073143.","mla":"Golmakani, Ali, et al. “Rigorous Numerics for Critical Orbits in the Quadratic Family.” <i>Chaos</i>, vol. 30, no. 7, 073143, AIP, 2020, doi:<a href=\"https://doi.org/10.1063/5.0012822\">10.1063/5.0012822</a>.","short":"A. Golmakani, E. Koudjinan, S. Luzzatto, P. Pilarczyk, Chaos 30 (2020)."},"issue":"7"},{"citation":{"ieee":"L. Chierchia and E. Koudjinan, “V. I. Arnold’s ‘pointwise’ KAM theorem,” <i>Regular and Chaotic Dynamics</i>, vol. 24. Springer, pp. 583–606, 2019.","chicago":"Chierchia, Luigi, and Edmond Koudjinan. “V. I. Arnold’s ‘Pointwise’ KAM Theorem.” <i>Regular and Chaotic Dynamics</i>. Springer, 2019. <a href=\"https://doi.org/10.1134/S1560354719060017\">https://doi.org/10.1134/S1560354719060017</a>.","ama":"Chierchia L, Koudjinan E. V. I. Arnold’s “pointwise” KAM theorem. <i>Regular and Chaotic Dynamics</i>. 2019;24:583–606. doi:<a href=\"https://doi.org/10.1134/S1560354719060017\">10.1134/S1560354719060017</a>","mla":"Chierchia, Luigi, and Edmond Koudjinan. “V. I. Arnold’s ‘Pointwise’ KAM Theorem.” <i>Regular and Chaotic Dynamics</i>, vol. 24, Springer, 2019, pp. 583–606, doi:<a href=\"https://doi.org/10.1134/S1560354719060017\">10.1134/S1560354719060017</a>.","ista":"Chierchia L, Koudjinan E. 2019. V. I. Arnold’s “pointwise” KAM theorem. Regular and Chaotic Dynamics. 24, 583–606.","short":"L. Chierchia, E. Koudjinan, Regular and Chaotic Dynamics 24 (2019) 583–606.","apa":"Chierchia, L., &#38; Koudjinan, E. (2019). V. I. Arnold’s “pointwise” KAM theorem. <i>Regular and Chaotic Dynamics</i>. Springer. <a href=\"https://doi.org/10.1134/S1560354719060017\">https://doi.org/10.1134/S1560354719060017</a>"},"arxiv":1,"publication":"Regular and Chaotic Dynamics","title":"V. I. Arnold’s “pointwise” KAM theorem","external_id":{"arxiv":["1908.02523"]},"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1908.02523"}],"status":"public","date_updated":"2021-01-12T08:20:34Z","day":"10","oa_version":"Preprint","page":"583–606","publication_status":"published","abstract":[{"text":"We review V. I. Arnold’s 1963 celebrated paper [1] Proof of A. N. Kolmogorov’s Theorem on the Conservation of Conditionally Periodic Motions with a Small Variation in the Hamiltonian, and prove that, optimising Arnold’s scheme, one can get “sharp” asymptotic quantitative conditions (as ε → 0, ε being the strength of the perturbation). All constants involved are explicitly computed.","lang":"eng"}],"extern":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","article_processing_charge":"No","intvolume":"        24","date_created":"2020-10-21T15:25:45Z","volume":24,"article_type":"original","month":"12","quality_controlled":"1","type":"journal_article","doi":"10.1134/S1560354719060017","language":[{"iso":"eng"}],"year":"2019","publisher":"Springer","date_published":"2019-12-10T00:00:00Z","author":[{"last_name":"Chierchia","full_name":"Chierchia, Luigi","first_name":"Luigi"},{"first_name":"Edmond","full_name":"Koudjinan, Edmond","orcid":"0000-0003-2640-4049","id":"52DF3E68-AEFA-11EA-95A4-124A3DDC885E","last_name":"Koudjinan"}],"_id":"8693","oa":1}]